Clary wrote: Am I alone in thinking the usual, acceptable way of saying this is 'I found the area [where] I live gets power cuts' (as a usual occurrence) or 'We had a power cut in the area/where I live.' ?
No. But ESL teaching is to work with what's given as much as to simply give what a native would say.
The intuition of the native is explained through the illustration of the awkwardness.
Giving OristarA a native-sounding phrase would be to give him a fish. Working with the sentences he came up with, even though they might be awkward, is to help teach him to fish.
Despite the position of some, it's not a competition to come up with the "best" sentence. I think we can best help OristarA by teaching the underlying reasons, as we are not going to be around to pick all his sentences for him.
OristarA's original question was about whether the sentences were correct. He went on to learn that they were not and he settled on a substitute that was unfortunately also incorrect.
IMHO, we should have as our focus helping him improve his English. And my personal methodology involves using what the ESL learner gives to work with, even if it's easier to just rewrite it for them.